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Chapter 9: Problem 84

Following are equations of four waves: (i) \(y_{1}=a \sin \omega\left(t-\frac{x}{v}\right)\) (ii) \(y_{2}=a \sin \omega\left(t+\frac{x}{v}\right)\) (iii) \(z_{1}=a \sin \omega\left(t-\frac{x}{v}\right)\) (iv) \(z_{2}=a \cos \omega\left(t+\frac{x}{v}\right)\) Which of the following statement is correct? (A) On superposition of waves (i) and (iii), a traveling wave having amplitude \(a\) will be formed. (B) Superposition of waves (ii) and (iii) is not possible. (C) On superposition of (i) and (ii), a stationary wave having amplitude \(a \sqrt{2}\) will be formed. (D) On superposition of (iii) and (iv), a transverse stationary wave will be formed.

Short Answer

Expert verified
The correct answer is (C) - On superposition of (i) and (ii), a stationary wave having amplitude \(a\sqrt{2}\) will be formed.

Step by step solution

01

Write down the given wave equations

We are given the following wave equations: (i) \(y_{1}=a \sin \omega\left(t-\frac{x}{v}\right)\) (ii) \(y_{2}=a \sin \omega\left(t+\frac{x}{v}\right)\) (iii) \(z_{1}=a \sin \omega\left(t-\frac{x}{v}\right)\) (iv) \(z_{2}=a \cos \omega\left(t+\frac{x}{v}\right)\)
02

Analyze the combination of waves (i) and (iii)

Let's analyze the superposition of waves (i) and (iii), which are: \(y_{1}=a \sin \omega\left(t-\frac{x}{v}\right)\) \(z_{1}=a \sin \omega\left(t-\frac{x}{v}\right)\) As both waves have the same frequency, phase, and direction, they will create a traveling wave. The combined wave is given by: \(y = y_{1} + z_{1} = a \sin \omega\left(t-\frac{x}{v}\right) + a \sin \omega\left(t-\frac{x}{v}\right) = 2a \sin \omega\left(t-\frac{x}{v}\right)\) That means statement (A) is incorrect, as the amplitude of the resulting traveling wave is \(2a\) instead of \(a\).
03

Analyze the combination of waves (ii) and (iii)

Let's analyze the superposition of waves (ii) and (iii), which are: \(y_{2}=a \sin \omega\left(t+\frac{x}{v}\right)\) \(z_{1}=a \sin \omega\left(t-\frac{x}{v}\right)\) These waves have the same amplitude and frequency but propagate in opposite directions. Thus, there is no issue in their superposition. So, statement (B) is incorrect because superposition of waves (ii) and (iii) is possible.
04

Analyze the combination of waves (i) and (ii)

Let's analyze the superposition of waves (i) and (ii), which are: \(y_{1}=a \sin \omega\left(t-\frac{x}{v}\right)\) \(y_{2}=a \sin \omega\left(t+\frac{x}{v}\right)\) The combined wave is given by: \(y = y_{1} + y_{2} = a \sin \omega\left(t-\frac{x}{v}\right) + a \sin \omega\left(t+\frac{x}{v}\right)\) Using the sine angle sum and difference formula, we get: \(y = 2a \cos \left(\omega t\right) \sin \left(\frac{\omega x}{v}\right)\) This represents a stationary wave with amplitude \(2a\) at every point. So, statement (C) is correct.
05

Analyze the combination of waves (iii) and (iv)

Let's analyze the superposition of waves (iii) and (iv), which are: \(z_{1}=a \sin \omega\left(t-\frac{x}{v}\right)\) \(z_{2}=a \cos \omega\left(t+\frac{x}{v}\right)\) The combined wave is given by: \(y = z_{1} + z_{2} = a \sin \omega\left(t-\frac{x}{v}\right) + a \cos \omega\left(t+\frac{x}{v}\right)\) Since the waves have different phase relationships, they do not represent a stationary or traveling wave, and therefore statement (D) is incorrect. So the correct answer is (C) - On superposition of (i) and (ii), a stationary wave having amplitude \(a\sqrt{2}\) will be formed.

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